*(note: this post has been closed to comments; comments about it on other pages will be deleted!)*

**UPDATE!!:** The saga continues at this post.

**MORE UPDATES, WITH REFUTATIONS!**

Every year I get a few kids in my classes who argue with me on this. And there are arguers all over the web. And I just know I'm going to get contentious "but it just *can't* be true" whiners in my comments. But I feel obliged to step into this fray.

.9 repeating equals one. In other words, .9999999... is the same number as 1. They're 2 different ways of writing the same number. Kind of like 1.5, 1 1/2, 3/2, and 99/66. All the same. I know some of you still don't believe me, so let me say it loudly:

Do you believe it yet? Well, I do have a couple of arguments besides mere size. Let's look at some reasons why it's true. Then we'll look at some reasons why it's not false, which is something different entirely. The standard algebra proof (which, if you modify it a little, works to convert any repeating decimal into a fraction) runs something like this. Let x = .9999999..., and then multiply both sides by 10, so you get 10x = 9.9999999... because multiplying by 10 just moves the decimal point to the right. Then stack those two equations and subtract them (this is a legal move because you're subtracting the same quantity from the left side, where it's called x, as from the right, where it's called .9999999..., but they're the same because they're equal. We said so, remember?):

Surely if 9x = 9, then x = 1. But since x also equals .9999999... we get that .9999999... = 1. The algebra is impeccable.

But I know that this is unconvincing to many people. So here's another argument. Most people who have trouble with this fact oddly *don't* have trouble with the fact that 1/3 = .3333333... . Well, consider the following addition of equations then:

This seems simplistic, but it's very, very convincing, isn't it? Or try it with some other denominator:

Which works out very nicely. Or even:

It will work for any two fractions that have a repeating decimal representation and that add up to 1.

Those are my first two demonstrations that our fact is true (the last one is at the end). But then the whiners start in about all the reasons they think it's false. So here's why it's not false:

- ".9 repeating doesn't equal 1, it gets closer and closer to 1."

May I remind you that .9 repeating is a *number*. That means it has it's place on the number line somewhere. Which means that it's not "getting" anywhere. It doesn't move. It either equals 1 or it doesn't (it does of course), but it doesn't "get" closer to 1.

- ".9 repeating is obviously less than 1."

Hmmmm...it might be obvious to you, but it's not obvious to me. Is it really less than 1? How much less than 1? No, seriously...tell me how much less? What is 1 minus .99999999...?

Really???? *Infinitely many* zeros and then after the *infinite* list that *never ends*, there's a 1???? Surely that's stranger than the possibility that .9 repeating simply does equal 1. Or for something even stranger, consider this: if .9 repeating is less than 1, then we ought to be able to do something very simple with those two numbers: find their average. What's the number directly between the two? Or for that matter, name *any* number between the two. Let me guess: the average is .99999...05? So after this *infinite* list of 9s, there's the possibility of starting up multiple-digit extensions? Doesn't that just raise the obvious question: What about .9999999...9999999...? Namely, infinitely many 9s, and then after that infinite list, there's *another* infinite list of 9s? How, exactly is that different from the original infinite list of 9s? If you saw it written out, where would the break between the lists be?

I'm afraid that if you apply the "huh??" test of strangeness, you get a much higher strangeness factor if you say that .9999999... is *not* 1 than you do if you say it *is* 1.

- "Uhhhhh, I'm sorry, but I still don't believe you. .99999... just can't equal 1."

Well, let's look a little more carefully at what we really mean by .999999...:

This equation isn't really up for debate, right? It's simply the meaning of our place value system made explicit. That thing on the right hand side is called an infinite geometic series. They have been studied extensively in math. The word "geometric" means that each term of the series is the identical multiple (in this case 1/10) of the previous term. The **definition** of the sum of an infinite geometric series (and other series, too, but we won't get into those) goes something this:

- Start making a list of partial sums: the sum of the first one number, then the sum of the first two numbers, then the sum of the first three, etc.
- Examine your list closely. In this case the list is: .9, .99, .999, .9999, .... (Note that the actual number .99999.... is not on the list, since every number on the list has finitely many 9s.)
- Find some numbers that are bigger than every single number on your list. Like 53, 3.14, and a million.
- Of all the numbers that are bigger than every number on your list, find the smallest possible such number. I think we can all agree that this smallest number is 1.
- That smallest number that can't be exceeded by anything on the list is the
**definition**of the sum of the geometric series.

Notice that I keep putting the word **definition** in bold face. (See, I did it again!) That's because it's a **definition**, which isn't really up for debate. It is the nature of a mathematical definition that once you acccept it, you have to agree to its consequences. In other words, .99999... = 1 by the **definition** of the sum of a geometric series. It's also true if you use the popular formula

a/(1 - r) with a = 9/10, and r = 1/10.

We're left with this: merely *saying* ".99999... doesn't equal 1" admits the fact that this number .99999... exists. And if it exists, it equals 1 by definition. The only way out for you now, if you still don't believe it, is to have a different working definition of the sum of an infinite series (go talk to some math professors, and see how far you get) or to deny the very existence of the number .9999.... I have seen a lot of people doubt that the number equals 1, but very few of them are willing to deny the very existence of that number. If you want to play "there's no such thing as infinitely long decimal representations," I'm afraid you won't get very far, because there's always the number pi to worry about, too, you know.

Okay, so there's my rant. .9 repeating equals one. No, I'm sorry, it does.

read through the and finally page... and noticed the 1/3 *3 giving 1 + 3E idea.

Though this may sound a little odd I have an explanation for this. You are claiming that the little bit left over in the 1/3 is equal to 1/oo (or E). heres the semi crazy part... its actually a third of E. I know it doesn't make sense right away. But if you remember it is a process then your remainder isn't an end point then you realize the next step is to divide that by three. Same as 2/3 being 2E/3. hence when you combine them you get 1 or multiply the 1/3 by 3. note that I am not saying that there is a number E/3, just a process.

Posted by: Brent K | February 29, 2008 at 05:25 PM

Brent K wrote:

[10x = 9.9999... is not equal because

0.9999....*10 != 9.9999...

Merely shifting the decimal point, like you assumed,is actually a short cut because 99% of the time trailing zeros are of no significance.

0.9999...*10 = 9.9999.. ..0

To make this clearer multiply any number by ten and keep the same number of decimal places. In essence, what happened was you assumed 0.9999..was a finite number or saying that infinity * 10 = infinity (as in the same number) When in actuality the second infinity is 10 times bigger then the first]

Brent, let me ask you the following very simple yes/no questions (feel free to elaborate if need be):

According to what you gave above, does

(1) 9 + 0.999... != 10 x 0.999... ?

(2) 3 + 0.333... != 10 x 0.333... ?

(3) 3 + 0.333... = 3.333... ?

(4) 10 x 0.333... = 3.333...0 ?

(5) 1000 x 0.00333... = 3.333...000 ?

(6) 10 x 0.333... != 1000 x 0.00333... ?

(7) 3.333... / 10 = 0.333...3 ? (/ means divided by)

(8) 10 x 0.333...3 = 3.333...0 ?

(9) 3.333... != 3.333...0 ?

(10) 10 x (3.333... / 10) != 3.333... ?

Hmmm, that last one (#10) may be a bit of a trick question depending on how you answer questions #4, #7, #8, and #9. Expect to be asked for more details about your reasoning once you give your answers.

Posted by: Monimonika | February 29, 2008 at 08:54 PM

Brent wrote:

[The reason you will not find a number between 1 and 0.999... is that it is the infinitely smallest distance away. And are therefore beside each other on the number line. Using this same reasoning I could ask you to find a integer between 2 and 3, and if you couldn't then they would be equal, then use deductive reasoning prove that x=y.]

Let me illustrate to you how your argument here is flawed:

Some Simple Math Statement (SSMS): "When any EVEN integer is divided by 2, it will yield an integer as a result."

pseudo-Brent: "Using this same reasoning I could ask you to divide the integer 3 by 2 and try to get an integer as a result." (pseudo-Brent fully implies that SSMS is flawed because of this.)

Brent, can you see what was wrong with the above argument by pseudo-Brent and how it relates to what you wrote?

Hint #1 (in case you somehow don't get it): 3 is not an even integer.

Hint #2: SSMS only makes a claim about a property of the even integers.

Hint #3: The set of integers is not the same as the set of reals.

Hint #4: The Density Property is not applied to the set of integers, but is applied to the set of reals. (refer back to Hint #3)

How does applying the Density Property to the set of integers (which it does not claim to apply to in the first place) help your case here? There's a word for something like this, it's called "non sequitur".

I suggest looking up the differences between integers and real numbers before tackling this issue again. It's one thing to have questions or slight misconceptions of the other stuff you wrote, but this one part was just sad.

Posted by: Monimonika | February 29, 2008 at 09:23 PM

Oh, by the way, the SSMS in the above post is assumed to be applied to even integers in base-10.

Posted by: Monimonika | March 01, 2008 at 09:29 AM

Before I begin answering your yes or no questions, let me say this, all the answers depend on whether you are keeping track of parts of infinity or not I will assume not for the answers.

1 no

2 no

3 yes

4 yes

5 yes

6 yes

7 yes, by this I mean if you hold your same amount of decimal places, if the end 3 is added then you have created a larger infinity(# of decimal places) then the first infinity and redefined a "new" infinity voiding the first.

8 yes, again assuming equal decimal places

9 not when the decimal places are equal

10 nope not when your infinity is held constant and you do the order of operations properly, 0.333...0

okay so I am seeing a little misunderstanding here, you are thinking that I am adding a digit after your infinite string of 3's when I am actually replacing the last digit, as to maintain equal amount of decimal

places therefore maintaining your infinite amount of decimal places. If you add a number to the end you have effectively created an infinity

plus one situation there by creating a new version of infinity.

Now for number ten think of it as an infinite array and you fill it with an infinite amount of 3s then you say that they all have to shift one space right to make space for the significant zero you want to add at the front. What happens is you either delete the last number or say its full and don't allow the operation(ie an overflow error)or the only other way I know of would be to keep this last digit stored somewhere else,like variable saying that there is 91E/30 left(which by the way you say is nothing because 1/infinity=0 0*anything=0, interesting paradox) , which is the argument I made for 1/3 *3 = 1, without this it would be E short. Now you go to take away the zero like a stack(ie all the other elements shift back the last space won't be 3, unless you can reference an outside variable (or had it set as a default,lol).If you know of another way to add the information while maintaining your original arrays size(even if it is "infinite") I would love to know,also I am sure there are a lot of people who would pay a lot of money to know how.

As to the part that is just sad,lol. I see what you mean by trying to use a property for things it was not intended for and you are correct in that. But it does follow as I was trying use integers as similarity (a simpler example) to try to show you what you are doing, but that obviously didn't work. I was trying to show you that saying that a number that is E away is the smallest step(even though it is supposed to be unquantifiable by definition) that you can make in your system is like 1 being the smallest step in the integer system. Thus showing that if you allow one number to equal another you end up allowing any number to equal any other given enough steps.

Since I answered your questions, I would like you to answer mine. Sorry mine can't be put in the yes/no simplicity category (unless you want to count binary), By the way my last question IS a trick question.

1)Find me a number that lies between any two adjacent numbers of the following list: 0.9999...8,0.999...7,0.999....6,

0.999...5,0.999...4,0.999...3,

0.999...2,0.999...1,0.999...90,

0.999...89,0.999...88 etc.

(note all have equal (whatever size of infinity you want) decimal places)

2)What is 1.333... - 1.000...?

( - means subtract by.

sub·trac·tion (sb-trkshn)

n.

The arithmetic operation of finding the difference between two quantities or numbers

dif·fer·ence (dfr-ns, dfrns)

n.

The amount by which one quantity is greater or less than another

Although I know that you already knew that,I was just showing you what a cheap shot you took)

(think hard is it 0.333... or infinity? or both since infinity can really represent any number depending on the circumstance)

if 0.333... did you take 1/0.333...(or 333...) steps to get there? if so then how big is was each step?

4)if 1 is no different(=) then 0.999..., why is it different then (!=)4

5)quantify infinity?

What it reduces to is that infinity is a concept and therefore any number containing ... is as good as saying infinity(/,*,-,+ )something = something.

Though I must say the paradox that came up is quite interesting, as in you need fractions of your smallest element to show that it is continuous, which by having smallest elements makes it not continuous.

Posted by: Brent K | March 01, 2008 at 03:46 PM

oh one more question. what happened to the zero that should be the last digit in your 0.999....*10 sequence?

Posted by: Brent K | March 01, 2008 at 03:52 PM

update. Density property with infinitesimal differences.

first lets start with r=0.999.. and s =1 and I will even do this your way and then show an easy way. :),Monimonika

r<(r+s)/2< s where r< s which is in fact what you are asking to show

to make it weighted you will agree that I can

use the following:

¨ (T) If x < y and y < z, then x < z.

¨ (A) If x < y and z is any real number, then z + x < z + y.

¨ (M) If x < y and z is any positive real number, then zx < zy.

(ar+br)/(a+b)<(ar+bs)/(a+b)<(as+bs)/(a+b)

¨ We have br < bs by (M), with x = r, y = s, z = b.

¨ Then ar + br < ar + bs by (A) with x = br, y = bs, z = ar.

¨ then

(ar+br)/(a+b)<(ar+bs)/(a+b) by (M) with x = ar + br, y = ar + bs and z= 1/(a+b)

now with the equation that allows for infinitely more numbers within 2 real numbers

(ar+br)/(a+b)<(ar+bs)/(a+b)<(as+bs)/(a+b)

lets say we make a = 10 and b = 2 to make it easy

this gives us

(10*0.999...+2*0.999...)/(10+2)<(10*0.999...+2*1)/(10+2)<(10*1+2*1)/(10+2)

(12*0.999...)/12<(9.999...0+2)/12<(12*1)/12

11.999...88/12<11.999.../12<12/12

0.999...< ? < 1

my question is what is the ?

11.999.../12

if we go by what you say, its 1

so lets write this out

0.999...<1<1

I don't know about you but 1<1 strikes me as making that argument false

okay... so then it must equal 0.999...

0.999...<0.999...<1

hmm... still false

so obviously the answer is between them, that it is 0.999...1666...

but that is tacking infinitely more numbers, effectively giving you infinity*2 decimal numbers to defeating your old infinity... interesting

now the easiest way of proving this false

hint:

in the hypothesis its stated that r<(r+s)/2< s where r< s

so now this is where pseudo Monimonika says "How does applying the Density Property to the set of integers (which it does not claim to apply to in the first place) help your case here? There's a word for something like this, it's called "non sequitur"."

well it follows like this you have done the exact same thing with real numbers.

if the proof had stated that r<=(r+s)/2<=s where r<=s it would have been correct... but also include any number ..shame

thereby stating that r < s, r=0.999... , s=1 and 0.999...=1 means that you are applying it to something it didn't claim to apply to in the first place(ie being less and equal at the same time).

have a good day:)

Posted by: Brent K | March 01, 2008 at 07:07 PM

On second thought sorry for being spiteful Monimonika. I shouldn't have stated it like that. Your posts attacked my intelligence, so I took it personally and retaliated.

Posted by: Brent K | March 01, 2008 at 07:22 PM

Thank you for answering the questions and clarifying your position on things. I will try to be clear on my part as well as try not to be condescending (as I admit I particularly was in my "that's sad" post. Apologies).

For the answers to my questions, let me first straighten out the equations to conform to your answers:

(*1) 9 + 0.999... = 10 x 9.999...

(*2) 3 + 0.333... = 10 x 3.333...

(*3) 3 + 0.333... = 3.333...

(*4) 10 x 0.333... = 3.333...0

(*5) 1000 x 0.00333... = 3.333...000

(*6) 10 x 0.333... != 1000 x 0.00333...

(*7) 3.333... / 10 = 0.333...3 (if holding same amount of decimal places)

(*8) 10 x 0.333...3 = 3.333...0 (assuming equal decimal places)

(*9) 3.333... = 3.333...0 (when decimal places are equal)

(*10) 10 x (3.333... / 10) = 3.333...

I admit that I do not follow your answer *2 (and *1). Yes, *2 is consistent with *3, *8, and *9, but how does simply adding 3 to 0.333... get you that "0" that you insist must come at the end of the infinite string of 3s in the result? There is no shift of the decimal place (the dot) going on, therefore no need to replace a digit to keep the infinite amount of decimal places the same. The same applies to *1. Without that "0" at the end, how can you claim that there is an infinitesimal amount of difference between "10 x 0.999..." and "0.999..." while also claiming *9 (which could be seen as saying 9.999... = 9.999...0)?

I guess the thing that really confuses me is: "Is 0.333... equal to 0.333...3 or 0.333...0?"

Please do not say that the above has to be defined for you at the very beginning. You seemed to be able to come up with your original argument without that bit of info just fine before.

If you say that "0.333... = 0.333...3", then:

3 + 0.333...3 = 3.333...3 (I see no reason to replace the final digit with a zero)

10 x 0.333...3 = 3.333...0

therefore

3 + 0.333...3 != 10 x 0.333...3

Contradicts *2, so "0.333... != 0.333...3".

If you say that "0.333... = 0.333...0":

0.999... = 0.999...0 (by multiplying both sides by 3)

9 + 0.999...0 = 10 x 0.999...0 (from *1)

9 + 0.999,,,90 = 10 x 0.999,,,90 (slight notation modification made-up to reveal the next-to-final decimal place digit. Think of the ",,," to be one decimal place less than "...")

9.999,,,90 = 9.999,,,00 (again, no reason for shifting of decimal place for the left side of the equation)

Wait, 9.999,,,90 can't be equal to 9.999,,,00, can it? There's a difference there! So, "0.333... != 0.333...0".

Both of my non-rigorous reasonings above are hinged on the assumption that addition does not shift the decimal places. Unless you can explain to me how addition is supposed to shift the decimal places, these stand as challenges to your argument of how multiplying by powers of 10 somehow necessitates placing zeros at the end due to decimal place shifting.

Brent wrote: [Now for number ten think of it as an infinite array and you fill it with an infinite amount of 3s then you say that they all have to shift one space right to make space for the significant zero you want to add at the front. What happens is you either delete the last number or say its full and don't allow the operation(ie an overflow error)or the only other way I know of would be to keep this last digit stored somewhere else,like variable saying that there is 91E/30 left(which by the way you say is nothing because 1/infinity=0 0*anything=0, interesting paradox) , which is the argument I made for 1/3 *3 = 1, without this it would be E short. Now you go to take away the zero like a stack(ie all the other elements shift back the last space won't be 3, unless you can reference an outside variable (or had it set as a default,lol).If you know of another way to add the information while maintaining your original arrays size(even if it is "infinite") I would love to know,also I am sure there are a lot of people who would pay a lot of money to know how.]

I'm not sure if you're making this my problem to solve, or if you're admitting that you really can't answer #10 without contradicting yourself. If it is the former, I suggest that you better switch your thinking over to the latter. I also have this feeling that the reason you are having problems is because you are limited by the finite capacity of your computer hardware, and thus can't handle the concept of actual infinity. To put it more simply, a cheap calculator would yield the result 0.6666667 for the operation 2/3 and 0.3333333 for 1/3. You are thinking exactly like that calculator, just with finitely more decimal places (which is still far, far, far less than infinite decimal places) and maybe without the rounding up. C'mon, you can think beyond what your computer hardware can, can't you?

And since when did I say that 1/infinity equals 0? The reals are defined with the Archimedean property, which states that there are no infinitely large or infinitely small elements. Therefore, there is no such number as "infinity" defined in the reals, nor are there non-zero infinitesimals. 1/infinity is utterly meaningless in the reals (0.000...1 does not exist in the reals). "Infinity" as a number and "infinite string of numbers" are NOT THE SAME THING.

Also, you obviously have not read up on what the set of reals are and what a real number is defined to be. It's like you redefining EVEN integers to include numbers that can be represented with 2n+1 (where n is a positive integer)(like 2*1+1=3) and then telling others they are wrong for not also doing so. Here's a link to the wikipedia article on that subject: http://en.wikipedia.org/wiki/Real_number

Try to understand what is meant by "continuous" and "uncountable". Please realize that you have misconceptions of what the reals are and stop applying those misconceptions to the reals.

If you haven't already done so, you might as well look at this other article as well (in fact, please do so): http://en.wikipedia.org/wiki/0.999...

Phew, now to your questions:

1) Wait, why do I have to do this? In case you misunderstood me, YOU (not me) are the one with the notion that you can tack on some other digit at the end of an infinite string of recurring digits (as in, 0.999...0) here. These "numbers" in your list are nonsensical in the reals, and my point was to show that your way of notation and thinking led to contradictions. Since I find the list to not contain any actual numbers, my answer would be "None". Also, I clarify my answer with the explanation that the Density Property does not apply here (I really shouldn't have to explain why by now), so there is no claim that the things in your list are equal to each other either.

2) According to what I know of the decimal system (base-10 reals), my answer is:

1.333... - 1.000... = 0.333...

Cheap shot? Are you referring to my "non-sequitur" jab, my "SSMS" analogy, or something else? I admit that the "non-sequitur" jab was a bit much and I could easily have done without it.

Brent wrote: [(think hard is it 0.333... or infinity? or both since infinity can really represent any number depending on the circumstance)]

Again, "infinity" as a number and "infinite string of numbers" are NOT THE SAME THING.

Brent wrote: [if 0.333... did you take 1/0.333...(or 333...) steps to get there? if so then how big is was each step?]

What unit of measurement are you using to measure the size of each of the steps? Here's what I did (*s are used to keep things aligned):

*1.333...

-1.000...

____________

*0

then

*1.333...

-1.000...

____________

*0.

then

*1.333...

-1.000...

____________

*0.3

then

*1.333...

-1.000...

____________

*0.33

then

*1.333...

-1.000...

____________

*0.333

then (since I understand that the "..."s indicate the same "3-0" pattern recurring with the same result of 3)

*1.333...

-1.000...

____________

*0.333...

(I also understand that 0.333... = 0.33333... = 0.333333333333333... etc., so don't be even try to claim that I "missed" or "skipped" decimal places. Remember, you're the one with the nonsensical "final digit" notation, not me.)

You count the number of steps I took and measure each of those steps' sizes however way you intended.

4) Brent wrote: [if 1 is no different(=) then 0.999..., why is it different then (!=)4]

Can you tell me what "then (!=)4" is supposed to mean? Or is the poor sentence structure supposed to be a jab at me on some place where I may not have made sense?

5) *dives for Wikipedia*

From http://en.wikipedia.org/wiki/Infinity :

[Infinity ... comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in philosophy, mathematics, and theology.

In mathematics, "infinity" is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is a different type of "number" than the real numbers. Infinity is related to limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals,[1] Russell's paradox, non-standard arithmetic, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite.]

Once again, "infinity" as a number and "infinite string of numbers" are NOT THE SAME THING.

I can see that I most likely didn't answer your question (#5) there, but I admit to not being sure what kind of answer would satisfy you. I could maybe look up synonyms for "infinite" from an online thesaurus and simply list them out, but that would seem a bit patronizing to you (I think).

Brent wrote: [What it reduces to is that infinity is a concept and therefore any number containing ... is as good as saying infinity(/,*,-,+ )something = something.

Though I must say the paradox that came up is quite interesting, as in you need fractions of your smallest element to show that it is continuous, which by having smallest elements makes it not continuous.]

Make sure to follow that link to the Wikipedia page about real numbers. You have a few misconceptions that need to be straightened out.

Brent wrote: [oh one more question. what happened to the zero that should be the last digit in your 0.999....*10 sequence?]

Umm, what zero? What "0.999....*10" sequence? I know that you're the one who wrote:

[0.9999...*10 = 9.9999.. ..0]

and that I merely just followed your lead for a bit in an attempt to show why putting that zero there is nonsensical. Please quote the part of my post that has this "sequence" you speak of.

Brent wrote: [Density property with infinitesimal differences.

first lets start with r=0.999.. and s =1 and I will even do this your way and then show an easy way. :),Monimonika]

Okay...

Brent wrote: [r<(r+s)/2< s where r< s which is in fact what you are asking to show

((bunch of stuff that don't seem to matter at all))

if we go by what you say, its 1

so lets write this out

0.999...<1<1

I don't know about you but 1<1 strikes me as making that argument false

okay... so then it must equal 0.999...

0.999...<0.999...<1

hmm... still false]

In other words, this just shows that "r = s" according to the Density Property, since "r < r<(r+s)/2< s.

Brent continues on: [so obviously the answer is between them, that it is 0.999...1666...

but that is tacking infinitely more numbers, effectively giving you infinity*2 decimal numbers to defeating your old infinity... interesting]

What? Seriously, why are you making random stuff up? Again, you're the one who is stuck on new/old infinities and whatnot, not me.

Brent wrote: [now the easiest way of proving this false

hint:

in the hypothesis its stated that r<(r+s)/2< s where r< s

so now this is where pseudo Monimonika says "How does applying the Density Property to the set of integers (which it does not claim to apply to in the first place) help your case here? There's a word for something like this, it's called "non sequitur"."

well it follows like this you have done the exact same thing with real numbers.

if the proof had stated that r<=(r+s)/2<=s where r<=s it would have been correct... but also include any number ..shame]

Brent, your blatant dishonesty is absolutely disgusting. The statement "r < (r+s)/2 < s" is true for finding a number between r and s IF "r < s". If r = s (which means "r < s" is false), then (r+s)/2 = r = s and vice-versa. Density Property is supported. 0.999... = 1. Where is the contradiction? Where? You fail to show that 1 != 0.999..., and on top of that you try to make it seem like I'M the one claiming that there must be another number between 0.999... and 1. You make me SICK. Your apology for your spitefulness is not accepted because I can tell that you are insincere to the core. Bye.

Posted by: Monimonika | March 02, 2008 at 05:58 AM

Dang typos and miss-pasted stuff:

[In other words, this just shows that "r = s" according to the Density Property, since "r < r<(r+s)/2< s.]

should be:

In other words, this just shows that "r = s" according to the Density Property, since "r <(r+s)/2 < s" is shown to be false.

Posted by: Monimonika | March 02, 2008 at 06:08 AM

let x = 0.9999....

=> 1-x = 1/(Inf.)

=> if 1=x then 1/(Inf.) = 0

are you with me so far?

Shall we use "your" algebra to elaborate 1/(Inf) = 0? What the heck, let's.

let y = (Inf.),

if 1/(Inf.) = 0 then 1/y = 0

shall we go ahead and multiply the equation through by y, i can't contain myself

(1*y)/y = 0*y

=> y/y = 0

=> 1 = 0

Woah, didn't see that comming.

Your problem is you're trying to use simple algebra which is out of place here. I'm not going to argue if it does or does Not equal to 1, but arguments like ".9 repeating is a number. That means it has it's place on the number line somewhere. Which means that it's not "getting" anywhere." really make me laugh. It is infinatelly close to 1, does that sound right to you?

Posted by: Alex | March 02, 2008 at 02:15 PM

There's a simple answer to this argument, which is that .9-repeating cannot actually exist as a number. Because infinity by definition goes on forever, it cannot exist, because everything in existence ends and everything in existence has a limit. .9-repeating is just a concept. It is a useful fiction we use as a reference. Therefore, it does not equal one, but it doesn't really not equal one either.

Posted by: Anonymous | March 08, 2008 at 09:57 PM

No, im sorry your [stupid].

it only works if fractions have repeating decimal equivalents not if it equals 1/4 or 1/2 etc.

oh and thanks for proving were god.

NEW RELIGION YAYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

(note: comment edited to delete offensive remark)

Posted by: Tribute | March 08, 2008 at 11:35 PM

I just had a question: is this vaild? Assuming that .999...=1

.9 = 1-.1

.99 = 1-.01

.999...= 1-.000...1

therefore wouldn't

1 = 1-.000...1

.000...1=0

???

We had a long, heated debate concerning infinity, .999 repeating and 1 in my IB Math class. My mind still hurts from thinking about this and logically I don't think .999 will ever equal one but mathematically it does. A girl told me that you cannot put a 'limit' on infinity (which makes sense) so my thinking is wrong but idk becuase if my "proof" is correct then mathematically .0000000000...1 =0

Thanks!

Posted by: CM | March 12, 2008 at 08:31 PM

CM,

Did you read the post all the way? Basically, Polymath's response is:

"Really???? Infinitely many zeros and then after the infinite list that never ends, there's a 1???? Surely that's stranger than the possibility that .9 repeating simply does equal 1."

You might also like to read the other posts he links to.

Posted by: Monimonika | March 13, 2008 at 02:30 PM

So then what about a function with an asymptote at 1. Are you saying that as it approaches 1 from the negative y-axis, it actually will equal 1 at infinity? If thats true then surely a function that approaches 0 from the positive y-axis to infinity will eventually equal 0. But I think neither is the case as .9 no matter what comes after will always have a tens value of nine. .999999 cant = 1 because .9 wont ever change.

Posted by: CM | March 16, 2008 at 11:20 PM

CM wrote:

[So then what about a function with an asymptote at 1. Are you saying that as it approaches 1 from the negative y-axis, it actually will equal 1 at infinity? If thats true then surely a function that approaches 0 from the positive y-axis to infinity will eventually equal 0.]

First, 0.999... is not a function. It's a number with an unchanging defined value. If you have a function that plots the progression of adding a 9 to the right of "0." (as in, 0.9, 0.99, 0.999, 0.9999, etc.) then yes, the function never does reach the asymptote 1.

CM wrote:

[But I think neither is the case as .9 no matter what comes after will always have a tens value of nine. .999999 cant = 1 because .9 wont ever change.]

You make the mistake of thinking that 0.999... is plotted somewhere on that function. It isn't. 0.999... is on 1, the asymptote. Every single plotted point on the function only has a finite number of 9s trailing to the right, so 0.999... (which has an infinite number of trailing 9s) cannot be on that function line. Yes, the function can approach the asymptote as it tacks on more 9s, but it cannot "reach" an infinite number of 9s (aka 0.999...) to come in contact with the asymptote 1 (aka 0.999...).

Besides, have you ever actually tried subtracting 0.999... from 1 to get this "difference" you talk about? And no, assuming that 0.999... is in the same set as {0.9, 0.99, 0.999, 0.9999, etc.} (which is the answer set of the function talked about above) just means you are already assuming that 0.999... =/= 1 before you even start.

This flawed assumption of yours leads you to think along the lines of:

("*"s are used to indicate blank space, not multiplication. It is best viewed copied and pasted into a text file in Notebook or equivalent, not a Word document, so that the characters line up.)

*1.000...0

-0.999...9

_________

The non-conventional notation above is there to illustrate your assumptions.

We know how to carry the one, right? I will use "T" to represent "10" as a single digit

*0.T00...0

-0.999...9

_________

Repeat

*0.9T0...0

-0.999...9

_________

And so on,

*0.999...T

-0.999...9

_________

-0.000...1

The above fits with your assumptions, right? But you had to replace 0.999... with this... "0.999...9" pseudo-number thingie to make it work. "0.999...9" apparently has an "infinite" number of trailing 9s, yet also ends at some point at which no more 9s can be added (thus being finite). Contradiction much?

Here's some actual subtraction that does not use your assumptions:

*1.000...

-0.999...

_________

Carrying the one:

*0.T00...

-0.999...

_________

Repeat:

*0.9T000...

-0.99999...

_________

Couple more repeats:

*0.999999999T00...

-0.999999999999...

__________________

Well, it's obvious that we would be carrying the one for all eternity (i.e. every single 0 to the right of "1." is changed to 9), so let's reflect that in our equation:

*0.999...

-0.999...

__________

Wait, notice something? The 1 just transformed into 0.999...! And since no subtraction has been performed yet (and carrying the one DOES NOT change the value either), 0.999... is equal in value to 1. Let's proceed with the actual subtraction:

*0.999...

-0.999...

_________

*0.000...

0.000... = 0

The difference is 0, as in, 0.999... = 1.

Posted by: Monimonika | March 18, 2008 at 10:21 AM

OK thanks. Youre argument seems valid and makes sense. One last question: are infinity and 0 reciprocals? (I know its off subject) Some kid "proved" they were, but if they are then infinity*0= 1 ???

Posted by: CM | March 18, 2008 at 07:09 PM

CM, I actually thank you back. Before I tried replying to your comments, I really did not understand what limits were and what all that other stuff actually meant. Researching a bit helped me finally get a clue to a certain degree. Thanks!

Anyway, to your question. There is a "proof" at this link: http://www.farid-hajji.net/fun/ge-1overinf.html

Okay, that was just a joke. But to seriously answer your question, you would have to understand that "infinity" is not a number, it's a concept. Therefore, you can't actually do simple things like addition, multiplication, division, subtraction, etc. to it (at least, not in the real numbers set). At this link:

http://mathforum.org/library/drmath/view/62486.html

Dr. Math gives this description of what is usually meant by "1/infinity = 0":

[In math, when you hear people say things like "1 over infinity is zero" what they are usually referring to is something called a limit. They are just using a kind of shorthand, however. They do NOT mean that 1 can actually be divided by infinity.

Instead, they mean that, if you divide 1 by successively higher numbers, the result becomes closer and closer to 0. If I divide 1 by a very large number, like a billion, then I get one-billionth, which is a VERY small number, but it isn't 0. Since there is no largest number, I can always divide 1 by a bigger number. But that will just produce an even smaller number, right? It will NEVER produce 0, no matter how high I go. But since the answer to the division is getting closer to and closer to 0, we say that "the limit of the expression is zero." But we have still not divided anything by infinity, since that isn't a number.]

This relates to the whole "function approaches but never reaches the asymptote" thing earlier. The asymptote is the limit. Since infinity is not an actual number that the function can calculate with, the function can never reach the limit/asymptote but can forever get closer to it.

Posted by: Monimonika | March 19, 2008 at 08:15 PM

Just to clarify, the whole "1/infinity = 0" thing really just means:

1/(a really big number) = (a really small number)

Thus,

(a really big number)*(a really small number) = 1

Zero is not involved here.

Posted by: Monimonika | March 19, 2008 at 08:59 PM

his first proof with 10x-x

endsu up having 9x=9

but if you start with .999999...

and multiply it by 9 you end up with 8.9999999999...

thus showing that the entire proof is just a rounding error and that 0 in this case is not just a place holder. it is an actual representation in the infinite

Posted by: E | March 22, 2008 at 09:46 PM

E,

Go up to four comments before yours to my comment on how to subtract 0.999... from 1.

Replace 1 and 0.999... with 9 and 8.999..., respectively.

Do the math and see if you can figure out why your "rounding error" claim is irrelevant as well as wrong. If you need clarification, don't hesitate to ask (but try to show that you actually read what I typed before doing so).

Posted by: Monimonika | March 24, 2008 at 07:15 PM

Well thanks for the help Monimonika. Infinity still racks my brian though... Just a question: where did you study? I want to study math and I need to start looking at universities.

Posted by: CM | March 25, 2008 at 09:26 PM

Amherst College

Amherst, MA

Went there mostly for the Japanese Department.

Posted by: Monimonika | March 26, 2008 at 12:31 PM

Sorry I did make a mistake on questions 1 and 2, I am guessing it was because of the not equals sign. I will get back to you on the rest of the counter statements, when I have more time. But I should say that I am finding myself to be in the constructivism philosophy of math category. After reading the last few posts, I am going to have to agree that 0.999.. is not a number because it is a process. Just as, it is not hard to understand that an infinitely long proof is impossible (ie. a proof that never ends, is not a proof).

Posted by: Brent | March 27, 2008 at 07:39 PM